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Glossary

C

Convergent (Series/Integral)

Criticality: 3

An infinite series or improper integral is convergent if its sum or value approaches a finite, real number. This means the sum or area under the curve does not grow infinitely large.

Example:

The improper integral ∫1∞1x2dx\int_1^{\infty} \frac{1}{x^2} dx is convergent because it evaluates to a finite value of 1.

D

Decreasing Function

Criticality: 2

A function $f(x)$ is decreasing over an interval if its values are continuously getting smaller as $x$ increases. This is a necessary condition for applying the Integral Test.

Example:

The function f(x)=1x2f(x) = \frac{1}{x^2} is a decreasing function on [1,∞)[1, \infty) because its derivative f′(x)=−2x3f'(x) = -\frac{2}{x^3} is always negative for x≥1x \ge 1.

Divergent (Series/Integral)

Criticality: 3

An infinite series or improper integral is divergent if its sum or value does not approach a finite number, often tending towards positive or negative infinity, or oscillating without limit.

Example:

The series ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} is divergent because its corresponding improper integral ∫1∞1xdx\int_1^{\infty} \frac{1}{\sqrt{x}} dx evaluates to infinity.

I

Improper Integral

Criticality: 3

An integral where at least one of the limits of integration is infinite, or the integrand has a discontinuity within the interval of integration. These are evaluated using limits.

Example:

The integral ∫1∞1xdx\int_1^{\infty} \frac{1}{x} dx is an improper integral because its upper limit extends to infinity.

Indefinite Integral

Criticality: 2

The set of all antiderivatives of a function, represented by $\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$ and C is the constant of integration.

Example:

The indefinite integral of f(x)=cos⁡(x)f(x) = \cos(x) is sin⁡(x)+C\sin(x) + C, representing all functions whose derivative is cos⁡(x)\cos(x).

Infinite Series

Criticality: 3

The sum of the terms of an infinite sequence, typically written in summation notation, whose convergence or divergence is often determined by tests like the Integral Test.

Example:

The expression ∑n=0∞(13)n=1+13+19+…\sum_{n=0}^{\infty} (\frac{1}{3})^n = 1 + \frac{1}{3} + \frac{1}{9} + \dots represents an infinite series.

Integral Test for Convergence

Criticality: 3

A test used to determine if an infinite series converges or diverges by comparing it to an improper integral of a related function. It applies when the function is positive, continuous, and decreasing.

Example:

To determine if the series ∑n=1∞1n2+1\sum_{n=1}^{\infty} \frac{1}{n^2+1} converges, one can use the Integral Test for Convergence by evaluating the improper integral ∫1∞1x2+1dx\int_1^{\infty} \frac{1}{x^2+1} dx.

P

Positive Function

Criticality: 2

A function $f(x)$ is positive over an interval if $f(x) > 0$ for all $x$ in that interval. This is a necessary condition for applying the Integral Test.

Example:

For the Integral Test, f(x)=e−xf(x) = e^{-x} is a positive function on [0,∞)[0, \infty) since all its values are greater than zero.

S

Sequence ($a_n$)

Criticality: 2

An ordered list of numbers, often defined by a formula, where each number corresponds to a positive integer index $n$. In the Integral Test, the terms of the series $a_n$ are derived from $f(n)$.

Example:

For the series ∑n=1∞nn2+1\sum_{n=1}^{\infty} \frac{n}{n^2+1}, the sequence is an=nn2+1a_n = \frac{n}{n^2+1}, generating terms like 12,25,310,…\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \dots.

U

U-Substitution

Criticality: 3

A technique for evaluating integrals by simplifying the integrand through a change of variables, often used when the integrand contains a composite function and its derivative.

Example:

To evaluate ∫xex2dx\int x e^{x^2} dx, one would use u-substitution by letting u=x2u = x^2, which transforms the integral into a simpler form.